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Finite-dimensional von Neumann entropy
Definition
Let be a nonzero finite-dimensional complex inner-product space, with the root convention that its inner product is linear in the first argument, and let be a density operator on (Density operator). If are its eigenvalues counted with multiplicity, define its von Neumann entropy by
The trace is the finite-dimensional endomorphism trace (The basis-independent trace of an endomorphism of a finite-dimensional vector space), and the spectral definition makes the expression independent of the eigenbasis. The term for a zero eigenvalue is defined to be . The default unit is nats, using the natural logarithm (The natural logarithm as the inverse of the exponential function); the entropy in bits is .
Depends on
- Density operator
- A non-negative operator is equivalently self-adjoint with nonnegative eigenvalues, a positive semidefinite matrix in an orthonormal basis, or an operator of the form S^*S
- The natural logarithm as the inverse of the exponential function
- The basis-independent trace of an endomorphism of a finite-dimensional vector space
Used by
- Complementary channels and coherent information Definition
- Finite classical-quantum states and Holevo information Definition
- Finite cq block codes, average error, and capacity Definition
- Finite IID quantum source and compression code Definition
- Finite quantum mutual, conditional, and conditional mutual information Definition
- Finite-dimensional quantum relative entropy Definition
- Operational quantum, private, and entanglement-assisted capacities Definition
- A Bell pair has negative quantum conditional entropy Example
- An orthogonal ensemble attains its Holevo information Example
- Entanglement-assisted capacity of a noiseless qubit Example
- Entropy of a diagonal qubit state Example
- Schumacher compression of a diagonal qubit source Example
- Strong subadditivity for a pure GHZ tripartite state Example
- Additivity of optimized entanglement-assisted information Lemma
- Coherent information equals output entropy minus environment entropy Lemma
- Coherent information of a qudit erasure channel Lemma
- Continuity of finite-dimensional quantum entropy Lemma
- Dimension-dependent quantum entropy continuity and Holevo control Lemma
- Entropy bounds for a finite mixture of states Lemma
- Entropy of a finite flagged state Lemma
- Exponential averaged spectral projectors for a finite cq ensemble Lemma
- Finite channel information maxima are attained Lemma
- The maximally mixed state maximizes finite quantum entropy Lemma
- Typical-projector bounds for finite quantum sources and cq ensembles Lemma
- Additive entanglement-assisted classical capacity formula Theorem
- Data processing for finite quantum mutual information Theorem
- Finite quantum mutual information identities and the Holevo identity Theorem
- Nonnegativity of finite-dimensional quantum conditional mutual information Theorem
- Regularized coherent-information formula for quantum capacity Theorem
- Regularized private-information formula for quantum channel capacity Theorem
- Strong subadditivity of finite-dimensional quantum entropy Theorem
Dependency tree · two levels
17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- John Watrous, The Theory of Quantum Information, §5.2.1 (standard reference, not scraped)
- Mark M. Wilde, From Classical to Quantum Shannon Theory, arXiv:1106.1445v8, §11.1 (standard reference, not scraped)